A Spectral Splitting Theorem for the $N$-Bakry Émery Ricci tensor
Wai-Ho Yeung
TL;DR
The paper generalizes spectral splitting to smooth metric measure spaces by proving that a complete non-compact weighted manifold with at least two ends and bounded $f$ splits isometrically as $\mathbb{R}\times X$ when $0<\gamma<\left(\frac{1}{(n-1)(1+(n-1)/N)}+\frac{n-1}{4}\right)^{-1}$ and $\lambda_1(-\gamma\Delta_f+\operatorname{Ric}^N_f)\ge0$. The approach combines $\mu$-bubble techniques with a detailed second-variation analysis of weighted bubbles, separating tangential, normal, and spectral contributions under the $N$-Bakry–Émery curvature framework to force $u$ to be constant and $\operatorname{Ric}^N_f\ge0$. This yields a splitting $M\cong\mathbb{R}\times X$ with $X$ endowed with nonnegative $N$-Bakry–Émery Ricci curvature, and the method recovers sharp constants known from Antonelli–Pozzetta–Xu and Catino–Mari–Mastrolia–Roncoroni. An alternative proof recovers the classical Ricci case by taking $f$ constant and demonstrates how the sharp spectral constant $4/(n-1)$ emerges in the limit $N\to0^+$, linking to the traditional Cheeger–Gromoll results.
Abstract
We extend the spectral generalization of the Cheeger-Gromoll splitting theorem to smooth metric measure space. We show that if a complete non-compact weighted Riemannian manifold $(M,g,e^{-f}\,dvolg)$ of dimension $n\ge 2$ has at least two ends where $f$ is smooth and bounded. If there is some $N\in (0,\infty)$ and $γ<\left(\frac{1}{(n-1)\left(1 + \frac{n-1}{N}\right)} + \frac{n-1}{4}\right)^{-1}$ such that $$λ_1(-γΔ_f+\operatorname{Ric}^N_f)\ge 0$$then $M$ splits isometrically as $\mathbb{R}\times X$ for some complete Riemannian manifold $X$ with $(\operatorname{Ric}_X)^N_f\ge 0$. The estimate can recover the spectral splitting result and its sharp constant $\frac{4}{n-1}$ in Antonelli-Pozzetta-Xu and and Catino--Mari--Mastrolia--Roncoroni.